Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{3}+11 x^{2}+39 x+29$$

Step-by-Step Solution

Answer

The roots of the polynomial are \(x = -1, -5, -5\) and the polynomial can be rewritten as a product of linear factors as \(f(x) = (x + 1)(x + 5)(x + 5)\). This can be verified graphically by plotting the polynomial and seeing where it intersects the x-axis.

Step 1: Identify the Polynomial

The given function is a polynomial \(f(x) = x^3 + 11x^2 + 39x + 29\). This is the equation whose roots need to be found.

Step 2: Find the roots

To find the roots of the polynomial, set \(f(x) = 0\), then solve for \(x\). This gives the equation \(x^3 + 11x^2 + 39x + 29 = 0.\) The roots of this equation can be found using the Rational Root Theorem, the details of which are complex and may require trial and error. However, the roots are \(x= -1, -5, -5\). If complex roots are present they can be found by using the quadratic formula on the remaining quadratic after factoring out the real root.

Step 3: Write as a product of linear factors

The polynomial can be written as a product of its linear factors using the roots just found. So, \(f(x) = x^3 + 11x^2 + 39x + 29 = (x + 1)(x + 5)(x + 5)\). In this form it is clear that the roots are indeed \(x = -1, -5, -5\).

Step 4: Verify these results graphically

Using a graphing utility, one can plot the polynomial \(f(x) = x^3 + 11x^2 + 39x + 29\). The function should intersect the x-axis at \(x = -1, -5, -5\), which would signify the roots of the polynomial. Please refer to the particular graphing utility's instructions on how to input and graph the polynomial.

Key Concepts

Linear Factors

Every polynomial can be expressed as a product of smaller polynomials, usually called factors. A linear factor is a polynomial factor of degree one, which means it is of the form \(x - r\), where \(r\) is a root of the polynomial. For instance, if you have a polynomial with roots \(x = -1\) and \(x = -5\), you can express the polynomial as \((x + 1)(x + 5)(x + 5)\). Here, each \(x + r\) is a linear factor.

• Linear factors are useful because they break down the polynomial into much smaller and simpler components.
• They also allow for easier identification of the polynomial's roots, as the zero of each linear factor corresponds directly to a root of the polynomial.
• This method is particularly helpful when trying to simplify complex polynomial equations.

By expressing polynomials as products of linear factors, you can quickly verify the roots by performing substitution back into the original equation to ensure they equate to zero.

Rational Root Theorem

The Rational Root Theorem is a tool used in algebra to provide a strategy for finding potential rational roots of a polynomial. It asserts that if a polynomial has any rational roots, they must be in the form \( \frac{p}{q} \), where \(p\) divides the constant term and \(q\) divides the leading coefficient. To apply this theorem:

• List all the factors of the constant term of the polynomial.
• List all the factors of the leading coefficient.
• Combine these factors to create possible rational roots.

For example, in the polynomial \(f(x) = x^3 + 11x^2 + 39x + 29\), the potential rational roots could be evaluated by checking factors of \(29\) and \(1\). Though it requires some trial and error, it narrows down potential rational roots to test and avoid unnecessary lengthy calculations.

Graphing Utility

A graphing utility is an invaluable tool for visualizing polynomials and confirming their roots visually. It can be software or a physical calculator with graphing capabilities.When graphing a polynomial, the points where the graph crosses the x-axis represent the real roots of the polynomial. Using the utility:

• Input the polynomial equation into the graphing function.
• Observe where the graph meets the x-axis to identify real roots.
• If instructed, check for complex roots, which might require further algebraic methods since they don’t appear directly on the graph.

For example, plotting \(f(x) = x^3 + 11x^2 + 39x + 29\) should show the graph crossing the x-axis at \(x = -1\) and \(x = -5\) among other possible points. This verification step is essential as it visually confirms calculations and offers insight into the behavior of polynomial functions.

Polynomial Roots

Polynomial roots are solutions or values of \(x\) that satisfy the equation \((f(x) = 0)\). Finding roots is central to solving polynomial equations as they provide critical points where the polynomial equals zero.Here are key takeaways on identifying polynomial roots:

• Roots can be real or complex.
• Real roots can be found exactly where the polynomial touches or crosses the x-axis.
• Complex roots, when present in pairs, can be found using algebraic methods like the quadratic formula or synthetic division if not immediately apparent.

For instance, the polynomial \(f(x) = x^3 + 11x^2 + 39x + 29\) has real roots at \(x = -1\) and \(x = -5\). Understanding the nature and behavior of polynomial roots is crucial in algebra as it extends into more complex functions and real-world applications.